A CHARACTERIZATION of FINITE ÉTALE MORPHISMS in TENSOR TRIANGULAR GEOMETRY Contents 1. Introduction 1 2. Strongly Separable

A CHARACTERIZATION OF FINITE ETALE´ MORPHISMS IN TENSOR TRIANGULAR GEOMETRY

BEREN SANDERS

Abstract. We provide a characterization of finite étalemorphisms in tensor triangular geometry. They are precisely those functors which have a conservative right adjoint, satisfy Grothendieck–Neeman duality, and for which the relative dualizing object is trivial (via a canonically-defined map).

Contents 1. Introduction1 2. Strongly separable algebras2 3. Separable algebras and triangulated categories 12 4. Finite ´etalemorphisms 13 5. Examples 15 References 20

1. Introduction The purpose of this note is to give a characterization of “finite étalemorphisms” in tensor triangular geometry. We follow the notation, terminology, and perspective of [BDS16]. In particular, we will work in the context of rigidly-compactly generated tensor-triangulated categories [BDS16, Def. 2.7]. The kind of characterization we have in mind is analogous to the following well-known characterization of smashing localizations: 1.1. Theorem. Smashing localizations of a rigidly-compactly generated tensor- triangulated category T are precisely those geometric functors f ∗ : T → S between rigidly-compactly generated tensor-triangulated categories whose right adjoint f∗ is fully faithful. We will recall a proof in Remark 3.9 below. Smashing localizations include, for example, restriction to a quasi-compact open subset of the Balmer spectrum. More generally, the tensor-triangular analogue of an étalemorphism is extension- of-scalars with respect to a commutative separable algebra (with smashing localizations being the special case of idempotent algebras). Finite étalemorphisms are, by definition, extension-of-scalars with respect to a compact commutative separable algebra (see Definition 4.1). Most smashing localizations are not finite étale morphisms, just as most open immersions are not proper. We will prove:

Date: June 30, 2021. Author is supported by NSF grant DMS-1903429. 1 2 BEREN SANDERS

1.2. Theorem. Finite étaleextensions of a rigidly-compactly generated tensor- triangulated category T are precisely those geometric functors f ∗ : T → S between rigidly-compactly generated tensor-triangulated categories which satisfy the following three properties: (a) f ∗ satisfies Grothendieck–Neeman duality; (b) the right adjoint f∗ is conservative; (c) the canonical map 1S → ωf is an isomorphism. The terminology and notation will be explained in Section4. We just remark that under hypothesis (a), the algebra f∗(1S) is rigid (a.k.a. dualizable) and hence has an associated trace map. This corresponds by adjunction to a canonical map 1S → ωf from the unit to the relative dualizing object, which hypothesis (c) asserts is an isomorphism. The keys to the theorem are the robust monadicity theorems which hold for triangulated categories and a deeper understanding of strongly separable algebras. Indeed, we begin the paper in Section2 with a treatment of strongly separable algebras in arbitrary symmetric monoidal categories which may be of independent interest. We prove, in particular, that a rigid commutative algebra is separable if and only if it is strongly separable if and only if its canonically-defined trace form is nondegenerate (Corollary 2.38). We then turn in Section3 to tensor-triangulated categories and the role separable algebras play in that setting. A key tool is a strengthened version of separable monadicity (Proposition 3.8). We define finite étalemorphisms and prove the main theorem (Theorem 4.7) in Section4. In Sec- tion5, we illustrate the theorem by giving some examples and non-examples of finite étalemorphisms in equivariant homotopy theory, algebraic geometry, and derived algebra. Acknowledgements : The author readily thanks Paul Balmer, Tobias Barthel, Ivo Dell’Ambrogio, Drew Heard and Amnon Neeman.

2. Strongly separable algebras We begin with a discussion of separable algebras in an arbitrary symmetric monoidal category. Although separable algebras are well-understood at this level of generality, we would like to clarify the notion of strongly separable algebra. Our main goal is to show that the equivalent characterizations of classical strongly separable algebras over fields established by [Agu00] have suitable generalizations to arbitrary symmetric monoidal categories. The main punch-line is that a rigid commutative algebra is separable if and only if it is strongly separable if and only if its trace form is nondegenerate (see Corollary 2.38). Moreover, this is the case if and only if it has the (necessarily unique) structure of a special symmetric Frobenius algebra. 2.1. Terminology. Throughout this section we work in a fixed symmetric monoidal category (C, ⊗, 1). The symmetry isomorphism will be denoted τ : A⊗B −→∼ B ⊗A. An object A in C is rigid (a.k.a. dualizable) if there exists an object DA such that DA⊗− is right adjoint to A⊗−. An algebra A is an associative unital monoid in C. The multiplication and unit maps will be denoted µ : A ⊗ A → A and u : 1 → A. 2.2. Definition. An algebra (A, µ, u) is separable if there exists a map σ : A → A⊗A such that FINITE ETALE´ MORPHISMS IN TENSOR TRIANGULAR GEOMETRY 3

(σ1) µ ◦ σ = idA, and (σ2) (1 ⊗ µ) ◦ (σ ⊗ 1) = σ ◦ µ = (µ ⊗ 1) ◦ (1 ⊗ σ) as maps A ⊗ A → A ⊗ A . In other words, A is separable if the multiplication map µ : A ⊗ A → A admits an (A, A)-bilinear section. 2.3. Remark. If we precompose such a section σ with the unit u : 1 → A, we obtain a map κ := σ ◦ u : 1 → A ⊗ A which satisfies (κ1) µ ◦ κ = u, and (κ2) (1 ⊗ µ) ◦ (κ ⊗ 1) = (µ ⊗ 1) ◦ (1 ⊗ κ) as maps A → A ⊗ A . Conversely, given such a κ, the map A → A ⊗ A displayed in( κ2) satisfies axioms (σ1) and( σ2). Thus, an algebra A is separable if and only if it admits a map κ : 1 → A ⊗ A satisfying( κ1) and( κ2). Such a map κ is called a separability idempotent. 2.4. Remark. We refer the reader to [AG60], [CHR65], [KO74], [Pie82, Chapter 10], and [For17] for further information about separable algebras and their role in classical representation theory and algebraic geometry. The following notion of a strongly separable algebra was originally studied by Kanzaki and Hattori [Hat65, Kan62]: 2.5. Definition. An algebra A is strongly separable if there exists a map κ : 1 → A⊗A satisfying( κ1),( κ2) and (κ3) κ = τ ◦ κ . In other words, A is strongly separable if it admits a symmetric separability idempotent. 2.6. Remark. For classical algebras over a field, [Agu00] provides several equivalent characterizations of strongly separable algebras. Our present goal is to clarify the extent to which these characterizations hold in an arbitrary symmetric monoidal category. For this purpose, the graphical calculus of string diagrams will be very convenient. We refer the reader to [Sel11, §§3–4] and [PS13, §2] for more information concerning these diagrams and suffice ourselves to remark that it is a routine exercise to convert a proof involving string diagrams into a detailed proof using commutative diagrams. 2.7. Notation. We’ll read our string diagrams from bottom to top. The multiplication map µ : A ⊗ A → A and the map κ : 1 → A ⊗ A will be represented by

µ and κ while the unit u : 1 → A and the identity id : A → A will be represented by

u = and idA =

Thus, for example, axiom( κ2) reads

(2.8) = 4 BEREN SANDERS

2.9. Proposition. An algebra (A, µ, u) is strongly separable if and only if there exists a morphism κ : 1 → A ⊗ A satisfying (κ2) and (κ4) µ ◦ τ ◦ κ = u . Proof. By deﬁnition an algebra is strongly separable if it admits a morphism κ satisfying( κ1),( κ2), and( κ3). It is immediate that( κ1) and( κ3) together imply( κ4). It is also immediate that( κ3) and( κ4) together imply( κ1). Thus, the claim will be established if we can prove that( κ2) and( κ4) together imply( κ3). Using Notation 2.7, observe:

(2.10) = = = = (unital) (κ4) (assoc) (κ2)

We can then rearrange the last diagram by pulling the left-hand multiplication to the right-hand side and continue:

(2.11) = = = = (κ2) (assoc) (κ4) (unital)

This establishes κ = τ ◦ κ which is axiom( κ3). 2.12. Corollary. Any commutative separable algebra is strongly separable. Proof. This follows from Proposition 2.9 since axiom( κ4) coincides with axiom( κ1) when the algebra is commutative. 2.13. Remark. For string diagrams involving a rigid object A, we’ll use the direction of a string to indicate whether it represents A or its dual DA. For example, the unit 1 → DA ⊗ A and counit A ⊗ DA → 1 are represented by

and respectively, and the unit-counit relations are given by

= and =

2.14. Deﬁnition. Let A be a rigid algebra in the symmetric monoidal category C. Its trace map tr : A → 1 is given by

1⊗η µ⊗1 (2.15) A ' A ⊗ 1 −−→ A ⊗ DA ⊗ A −−→1⊗τ A ⊗ A ⊗ DA −−−→ A ⊗ DA −→ 1 . FINITE ETALE´ MORPHISMS IN TENSOR TRIANGULAR GEOMETRY 5

2.16. Remark. To explain this deﬁnition, recall that every endomorphism f : A → A of the rigid object A has an associated “trace” Tr(f): 1 → 1 given as η 1⊗f 1 −→ DA ⊗ A −−−→ DA ⊗ A −→τ A ⊗ DA −→ 1 . Moreover, post-composition by the map (2.17) DA ⊗ A −→τ A ⊗ DA −→ 1 induces a function C(A, A) ' C(1,DA ⊗ A) → C(1, 1) which sends f to Tr(f). On the other hand, the multiplication map µ : A ⊗ A → A corresponds by adjunction to a morphism A → DA ⊗ A given by η⊗1 1⊗µ (2.18) A ' 1 ⊗ A −−→ DA ⊗ A ⊗ A −−→1⊗τ DA ⊗ A ⊗ A −−−→ DA ⊗ A. Post-composition by this map provides a function C(1,A) → C(1,DA ⊗ A) ' C(A, A) which sends a morphism a : 1 → A to “left multiplication by a”: 1⊗a τ µ La : A ' A ⊗ 1 −−→ A ⊗ A −→ A ⊗ A −→ A. The map (2.15) deﬁning the trace map tr : A → 1 is readily checked to equal the composite of (2.18) and (2.17). Post-composition by the trace map thus provides the function C(1,A) −−→tr∗ C(1, 1)

a 7−→ Tr(La) . Thus tr : A → 1 is morally the map which sends an “element” of A to the trace of left multiplication by that element. 2.19. Deﬁnition. The trace form of a rigid algebra A is the map t : A ⊗ A → 1 deﬁned as the composite µ A ⊗ A −→ A −→tr 1 . 2.20. Remark. The trace map and trace form of a rigid algebra are given by the following string diagrams:

tr = and t =

2.21. Remark. A map f : A ⊗ A → 1 is said to be an “invariant” form (also called an “associative” form) if f ◦ (µ ⊗ 1) = f ◦ (1 ⊗ µ). Note that any form which factors through µ (such as the trace form of a rigid algebra) is necessarily invariant by the associativity of the multiplication. The converse is also true: A form A ⊗ A → 1 is invariant if and only if it factors through µ. In fact, we obtain a bijection maps A → 1 −→∼ invariant forms A ⊗ A → 1 given by θ 7→ θ ◦ µ with inverse f 7→ f ◦ (u ⊗ 1) = f ◦ (1 ⊗ u). 6 BEREN SANDERS

2.22. Remark. A map f : A ⊗ A → 1 is said to be a “symmetric” form if f = f ◦ τ. Note that if A is a commutative algebra then every invariant form is automatically symmetric. 2.23. Proposition. The trace form of a rigid algebra is symmetric. Proof. First note that we can rewrite the trace form as follows

(2.24) =

as has already been mentioned in Remark 2.16. Next we establish

(2.25) =

which is an equality of morphisms A ⊗ A → DA ⊗ A. By adjunction this can be checked after applying A ⊗ − and post-composing with A ⊗ DA ⊗ A −−→⊗1 A:

======

Next note that:

(2.26) = =

Then

======(2.24) (2.25) (2.26) (2.25) (2.24)

establishes that the trace form is symmetric. 2.27. Remark. Intuition for why the trace form is symmetric comes from the fact that for any two endomorphisms f, g : A → A, we have Tr(f ◦g) = Tr(g◦f). Hence, at least morally, t(a, b) = Tr(Lab) = Tr(La ◦ Lb) = Tr(Lb ◦ La) = Tr(Lba) = t(b, a). FINITE ETALE´ MORPHISMS IN TENSOR TRIANGULAR GEOMETRY 7

2.28. Deﬁnition. If A is a rigid object in a symmetric monoidal category, then every map f : A ⊗ A → 1 gives rise to two morphisms A → DA by adjunction (moving each copy of A to the right-hand side). These two maps A → DA coincide when f is symmetric, and are given by η⊗1 1⊗f (2.29) f ∗ : A ' 1 ⊗ A −−→ DA ⊗ A ⊗ A −−−→ DA ⊗ 1 ' DA. We say that a symmetric form f : A ⊗ A → 1 is nondegenerate if f ∗ : A → DA is an isomorphism. 2.30. Proposition. The trace form of a strongly separable rigid algebra is nondegenerate. Moreover, a strongly separable rigid algebra has a unique symmetric separability idempotent, which is given by

η (t∗)−1⊗1 (2.31) 1 −→ DA ⊗ A −−−−−−→ A ⊗ A. Proof. Let κ be a symmetric separability idempotent. We’ll start by showing that the composite (2.32) A ' 1 ⊗ A −−−→κ⊗1 A ⊗ A ⊗ A −−→1⊗t A ⊗ 1 ' A is the identity, where t denotes the trace form (Def. 2.19). First note:

(2.33) = =

Then observe that

= = = (κ2) (κ2)

and

= = = = (2.33) (κ4)

which shows that (2.32) is the identity map. Now κ is symmetric by assumption and the trace form t is symmetric by Proposition 2.23. Hence:

t t t t = = =

κ κ κ κ 8 BEREN SANDERS

In other words, the composite (2.32) coincides with the other composite

A ' A ⊗ 1 −−−→1⊗κ A ⊗ A ⊗ A −−→t⊗1 1 ⊗ A ' A.

It follows that the map κ∗ : DA → A given by

DA ' 1 ⊗ DA −−−→κ⊗1 A ⊗ A ⊗ DA −−→1⊗ A ⊗ 1 ' A is an inverse to t∗ : A → DA. Indeed:

t t∗ κ∗ κ = = and = =

∗ ∗ t κ κ t

In particular, the trace form is nondegenerate. Moreover, one can readily check ∗ that (t ⊗ 1) ◦ κ = η from which it follows that κ is given by (2.31).

2.34. Theorem. A rigid algebra is strongly separable if and only if its trace form is nondegenerate.

Proof. The only if part is provided by Proposition 2.30. Conversely, suppose A is a rigid algebra whose trace form t : A ⊗ A → 1 is nondegenerate. Write θ : A −→∼ DA for the associated isomorphism (that is, θ = t∗ in the notation of Def. 2.28) and deﬁne κ : 1 → A ⊗ A by

η −1 (2.35) 1 −→ DA ⊗ A −−−−→θ ⊗1 A ⊗ A.

First we check that t and κ form a self-duality in the sense that

t θ−1 t θ−1 (2.36) = = = (2.35) (2.29) κ θ

and

t t θ (2.37) = = = (2.35) (2.29) κ θ−1 θ−1 FINITE ETALE´ MORPHISMS IN TENSOR TRIANGULAR GEOMETRY 9

Armed with this relationship between t and κ, the fact that t is symmetric (by Proposition 2.23) implies that κ is also symmetric:

t t

t = κ = κ = = = (2.36) (2.23) (2.36)

κ κ κ κ κ κ κ

This establishes axiom( κ3). Next we establish( κ2), visualized in string diagrams in (2.8). It suﬃces to check equality after post-composition by the isomorphism θ ⊗ idA. Then by adjunction it suﬃces to check equality after applying A ⊗ − and post-composing by A ⊗ DA ⊗ A −−→⊗1 A. Indeed

t t θ θ θ

= = = = = (2.35) (2.37) (†) (2.29) θ−1

κ κ κ κ where the equality (†) is the fact that the trace form is an invariant form (Re- mark 2.21). Finally, we establish( κ1). Observe that

= t = = = = (‡)

κ κ κ κ κ and note that we showed (‡) was a consequence of( κ2) in the proof of Proposi- tion 2.30. Precomposing with the unit we obtain

= =

κ κ which is( κ1). 2.38. Corollary. A rigid commutative algebra is separable if and only if it is strongly separable if and only if its trace form is nondegenerate. Proof. Every commutative separable algebra is strongly separable (Corollary 2.12) hence the claim follows from Theorem 2.34. 10 BEREN SANDERS

2.39. Remark. A rigid strongly separable algebra A is automatically self-dual, since the nondegeneracy of the trace form provides an isomorphism A =∼ DA. 2.40. Example. Consider the case where C = R - Mod is the category of R-modules for R a commutative ring. An R-algebra A is rigid precisely when it is finitely generated and projective (equivalently, finitely presented and flat) as an R-module. The trace map A → R is a 7→ Tr(La) where La : A → A denotes left multiplication by a, and the trace form t : A ⊗ A → R is given by t(a ⊗ b) = Tr(Lab). In this example, the argument in Remark 2.27 shows immediately that the trace form is symmetric. It turns out that over a field R = k, a separable algebra is automatically rigid (that is, finite-dimensional), as shown by [VZ66, Prop. 1.1]. It was partly to clarify such finiteness assumptions that led the author to write this section on strongly separable algebras in arbitrary symmetric monoidal categories. 2.41. Example. An idempotent algebra in a symmetric monoidal category is an algebra (A, µ, u) whose multiplication map µ : A ⊗ A → A is an isomorphism. This is equivalent to the equality u ⊗ A = A ⊗ u of morphisms A → A ⊗ A (which then serve as an inverse to µ). It is also equivalent to the switch map τ : A ⊗ A → A ⊗ A being equal to the identity map A ⊗ A → A ⊗ A. Idempotent algebras are thus examples of commutative (strongly) separable algebras. They have a (unique) separability idempotent given by µ−1 ◦u : 1 → A⊗A. However, they are usually not rigid. For example, take C = R - Mod for R a commutative ring. The idempotent R-algebra R[1/s] is rarely finitely generated as an R-module. Indeed, this would imply that the principal open D(s) ⊂ Spec(R) is both an open and closed subset of Spec(R); see the argument in [San19, Example 7.4], for example. 2.42. Remark. A discussion of separable algebras would not be complete without saying something about their relationship with Frobenius algebras: 2.43. Definition. A Frobenius algebra in a symmetric monoidal category is an object A equipped with both an algebra structure (A, µ, u) and a coalgebra structure (A, ∆, c) such that the Frobenius law holds: (1 ⊗ µ) ◦ (∆ ⊗ 1) = ∆ ◦ µ = (µ ⊗ 1) ◦ (1 ⊗ ∆). See, for example, [Koc04, 3.6.8]. We say that A is a symmetric Frobenius algebra if the invariant form µ A ⊗ A −→ A −→c 1 is symmetric. Thus, every commutative Frobenius algebra is symmetric. A special Frobenius algebra is a Frobenius algebra such that µ ◦ ∆ = idA. 2.44. Remark. If (A, µ, u, ∆, c) is a Frobenius algebra then the underlying object A is necessarily self-dual (cf. Rem. 2.39). Indeed, the two maps c ◦ µ : A ⊗ A → 1 and ∆ ◦ u : 1 → A ⊗ A provide a self-duality. 2.45. Remark. The following relationship between strongly separable algebras and special symmetric Frobenius algebras is well-known classically; we include a proof for precision and completeness. The interested reader will find more concerning these ideas in [LP07, Section 2.5], [FRS02, Section 3.3] and [Fau13], among other sources. 2.46. Proposition. An algebra admits the structure of a special symmetric Frobe- nius algebra if and only if it is rigid and strongly separable. In this case, the FINITE ETALE´ MORPHISMS IN TENSOR TRIANGULAR GEOMETRY 11 special symmetric Frobenius structure is unique: The counit A → 1 is the trace map (Def. 2.14) and the comultiplication A → A ⊗ A is the map corresponding (Rem. 2.3) to the unique symmetric separability idempotent (Prop. 2.30). Proof. If A is a strongly separable rigid algebra with (unique) symmetric separability idempotent κ : 1 → A ⊗ A then the corresponding map A → A ⊗ A is coassociative. Indeed, using both descriptions provided by( κ2) we have:

= = = = = (κ2) (κ2) (κ2) (κ2)

This provides A with the structure of a coalgebra with counit A → 1 given by the trace map. For the counital axiom just observe that

= = = and = = = (κ2) (†) (κ2) (†) where the last equalities (†) were established in the proof of Proposition 2.30. Al- ternatively, one can use the description (2.31) of the unique separability idempotent and check the counital diagrams after post-composition by the isomorphism t∗ : A −→∼ DA. This establishes that a strongly separable rigid algebra admits the structure of a special symmetric Frobenius algebra. Now suppose that (A, µ, u, ∆, c) is a special symmetric Frobenius algebra. Every Frobenius algebra is self-dual (Remark 2.44) and the comultiplication ∆ : A → A ⊗ A satisfies( σ2). In our case, it also satisfies( σ1) since A is assumed to be special. Symmetry of the associated separability idempotent ∆ ◦ u : 1 → A ⊗ A then follows from the assumed symmetry of c ◦ µ : A ⊗ A → A via the self-duality (as in the beginning of the proof of Theorem 2.34). Thus A is strongly separable with symmetric separability idempotent ∆ ◦ u. To establish uniqueness, first observe that the following commutative diagram

A u⊗1 A ⊗ A ∆⊗1 A ⊗ A ⊗ A

∆ 1⊗∆ 1⊗µ µ⊗1 A ⊗ A u⊗1⊗1 A ⊗ A ⊗ A A ⊗ A

id shows that the comultiplication ∆ of a Frobenius algebra is determined by ∆ ◦ u and µ. If (A, µ, u, ∆1, c1) and (A, µ, u, ∆2, c2) are two special symmetric Frobenius structures on the (rigid strongly separable) algebra (A, µ, u) then ∆1 ◦ u = ∆2 ◦ u by the uniqueness of symmetric separability idempotents (Prop. 2.30) and hence ∆1 = ∆2. Moreover, since ci ◦ µ : A ⊗ A → 1 and ∆i ◦ u : 1 → A ⊗ A form a self-duality for each i = 1, 2, we have:

1 1 2 2

= = . 1 = 2 12 BEREN SANDERS

That is, c1 ◦ µ = c2 ◦ µ. Precomposing by the unit we conclude that c1 = c2. This establishes that an algebra admits at most one special symmetric Frobenius structure. Finally, we have already proved that if A admits a special symmetric Frobenius structure then it is rigid and strongly separable and consequently the trace map and the symmetric separability idempotent provide it with a special symmetric Frobenius structure. These thus provide the unique such structure. 3. Separable algebras and triangulated categories In this section, we recall the relationship between separable algebras and tensor- triangulated categories established in [Bal11]. 3.1. Remark. Recall from [Bal11, Section 5] that for each 2 ≤ N ≤ ∞, there is the notion of an N-triangulated category (or triangulated category of order N) which includes as part of the structure a distinguished class of n-triangles for each n ≤ N which are required to satisfy suitable higher octahedral axioms. A 2-triangulated category is precisely the same thing as a pre-triangulated category, while the usual notion of triangulated category (in the sense of Verdier) lies between the notion of 2-triangulated and 3-triangulated. An N-triangulated functor is a functor which commutes with the suspension and preserves distinguished N-triangles (equivalently, preserves distinguished n-triangles for all n ≤ N). 3.2. Example. The homotopy category Ho(C) of a stable ∞-category has the structure of an ∞-triangulated category. 3.3. Remark. A tensor-triangulated category is a triangulated category equipped with a closed symmetric monoidal structure which is compatible with the triangulation in the sense of [HPS97, Definition A.2.1]. For 2 ≤ N ≤ ∞, we similarly have the notion of an N-tensor-triangulated category by replacing all instances of “triangulated” in the definition with “N-triangulated”. By an (N-)tensor-triangulated functor we mean an (N-)triangulated functor which is also a strong symmetric monoidal functor. 3.4. Example. The homotopy category Ho(C) of a presentably symmetric monoidal [NS17, Def. 2.1] stable ∞-category is an ∞-tensor-triangulated category. 3.5. Example. If A is a commutative separable algebra in an N-tensor-triangulated category T (2 ≤ N ≤ ∞) then the Eilenberg–Moore category A - ModT inherits the structure of an N-tensor-triangulated category such that the extension-of-scalars functor FA : T → A - ModT is an N-tensor-triangulated functor. The distinguished n-triangles in A - ModT (n ≤ N) are precisely those which are created by the forgetful functor UA : A - ModT → T. This is established by [Bal11, Theorem 5.17] and [Bal14, Section 1]. 3.6. Remark. The main theorem of [DS18] states that if T is an idempotent-complete triangulated category, then any triangulated adjunction F : T S : G is essentially monadic (that is, monadic up to idempotent completion and killing the kernel of G) whenever the Eilenberg–Moore category inherits a triangulation from T: \ ∼ (3.7) (S/ ker G) = GF - ModT . This theorem also holds (with the same proof) in the 2-category of N-triangulated categories for any 2 ≤ N ≤ ∞. In this case, the equivalence (3.7) is an equivalence of N-triangulated categories. To be clear, this is under the hypothesis FINITE ETALE´ MORPHISMS IN TENSOR TRIANGULAR GEOMETRY 13 that the Eilenberg–Moore category GF - ModT inherits an N-triangulation from the N-triangulation of T (see [DS18, Remark 1.8]). This is a strong hypothesis on the adjunction but, as established by Balmer (Example 3.5), does hold in the separable case. The following proposition clarifies the situation with the tensor: 3.8. Proposition. Let F : T → S be an (N-)tensor-triangulated functor with T idempotent complete. Suppose F admits a right adjoint G and that the F a G adjunction satisfies the right projection formula [BDS15, Definition 2.7]. If the commutative algebra G(1) ∈ T is separable then we have an induced equivalence \ ∼ (S/ ker G) = G(1) - ModT of (N-)tensor-triangulated categories. Proof. By [BDS15, Lemma 2.8], the projection formula implies that the monad of the adjunction is the monad associated to the algebra G(1). The separability of this monad implies that every s ∈ S is a direct summand of FG(s). It then follows from the projection formula that the thick subcategory ker G is a tensor-ideal. Thus S/ ker G and its idempotent completion (S/ ker G)\ inherit tensor-structures from S. On the other hand, the Kleisli category G(1) - FreeT inherits a tensor- structure from T such that the canonical functor T → G(1) - FreeT is a strict symmetric monoidal functor. The canonical functor K : G(1) - FreeT → S then inherits the structure of a strong symmetric monoidal functor from the corresponding structure on F . Since G(1) is separable, the Eilenberg–Moore category inherits a triangulation from T (Example 3.5) hence by [DS18] and Remark 3.6, we have equivalences \ ∼ \ ∼ (G(1) - FreeT) −→ (S/ ker G) −→ G(1) - ModT . The first functor is a strong symmetric monoidal equivalence. It follows that the second functor is also a symmetric monoidal equivalence since the tensor structure ∼ \ on G(1) - ModT = (G(1) - FreeT) is the idempotent completion of the tensor structure on the Kleisli category (see [Pau15, Section 1.1] and [Bal14, Section 1]). 3.9. Remark. As an application of the proposition, we can provide a proof of The- orem 1.1 stated in the Introduction which characterizes smashing localizations of rigidly-compactly generated categories. Proof of Theorem 1.1. The (⇒) direction is well-known: Any smashing localization of a rigidly-compactly generated category is a geometric functor to a rigidly- compactly generated category (whose right adjoint is fully faithful); see [HPS97, Section 3.3]. For the (⇐) direction, recall that smashing localizations are nothing but extension-of-scalars with respect to idempotent algebras. Suppose f ∗ : D → C is a geometric functor whose right adjoint f∗ is fully faithful. The multiplication map f∗(1C) ⊗ f∗(1C) → f∗(1C) becomes, under the projection formula, the counit ∗ ∗ f∗(): f∗f f∗(1C) ' f∗(1D ⊗ f f∗(1C)) ' f∗(1C) ⊗ f∗(1C) → f∗(1C). This is an isomorphism since f∗ is fully faithful. Thus f∗(1C) is an idempotent algebra. Idempotent algebras are separable, so Proposition 3.8 gives the result. 4. Finite etale´ morphisms The idea that extension-of-scalars with respect to a commutative separable algebra provides tensor triangular geometry with an analogue of an étaleextension goes back to the work of Balmer [Bal15, Bal16a, Bal16b]. Here we focus on finite étaleextensions of rigidly-compactly generated categories. 14 BEREN SANDERS

4.1. Definition. A geometric functor f ∗ : D → C between rigidly-compactly generated (N-)tensor-triangulated categories is finite étale if there exists a compact commutative separable algebra A in D and an (N-)tensor-triangulated equivalence1 ∼ ∗ C = A - ModD such that the functor f becomes isomorphic to the extension-of- scalars functor FA : D → A - ModD. 4.2. Remark. It follows from [Bal16a, Theorem 4.2] that if D is rigidly-compactly generated then A - ModD is also rigidly-compactly generated (for A a commutative separable algebra in D). Thus, there is no loss of generality in considering only geometric functors between rigidly-compactly generated categories. 4.3. Remark. Recall from [BDS16] that a geometric functor f ∗ : D → C between rigidly-compactly generated tensor-triangulated categories has a right ad- ! joint f∗ : C → D which itself has a right adjoint f : D → C. The relative dualizing ∗ ! ∗ object of f is the object ωf := f (1D) ∈ C. Recall that f is said to satisfy Grothendieck–Neeman duality if the right adjoint f∗ preserves compact objects. (A number of equivalent definitions are provided by [BDS16, Theorem 3.3].) In this case, the commutative algebra f∗(1C) is compact=rigid. Hence it has a trace map f∗(1C) → 1D (Def. 2.14), which corresponds to a map 1C → ωf .

4.4. Remark. In general, morphisms 1C → ωf can be identiﬁed with morphisms f∗(1C) → 1D by adjunction and these can be identiﬁed as in Remark 2.21 with the invariant forms on the algebra f∗(1C): ∼ ∼ 1C → ωf −→ f∗(1C) → 1D −→ invariant forms f∗(1C) ⊗ f∗(1C) → 1D .

Also recall (Def. 2.28) that an invariant form f∗(1C) ⊗ f∗(1C) → 1D is nondegenerate if the adjoint morphism f∗(1C) → Df∗(1C) is an isomorphism. (Note that these invariant forms are automatically symmetric since the algebra f∗(1C) is commutative.) On the other hand, recall from [BDS16, (2.18)] that we have an isomorphism f∗(ωf ) ' Df∗(1C).

4.5. Lemma. Let θ : 1C → ωf be any morphism. The map f∗(1C) → Df∗(1C) which is adjoint to the invariant form on f∗(1C) corresponding to θ coincides with the map

f∗(θ) (4.6) f∗(1C) −−−→ f∗(ωf ) ' Df∗(1C). Consequently, the invariant form associated to θ is nondegenerate if and only if f∗(θ) is an isomorphism. Proof. This is a straightforward veriﬁcation. From the deﬁnition of the isomorphism f∗(ωf ) ' Df∗(1C) = [f∗(1C), 1D] in [BDS16, (2.18)], one sees that the morphism (4.6) is obtained by going along the top of the following commutative diagram coev [1,lax] [1,] f∗(ωf ) [f∗(1C), f∗(ωf ) ⊗ f∗(1C)] [f∗(1C), f∗(ωf )] [f∗(1C), 1D] f∗(θ) [1,f∗(θ)] coev [1,lax] f∗(1C) [f∗(1C), f∗(1C) ⊗ f∗(1C)] [f∗(1C), f∗(1C)]

1For tensor-triangulated categories in the usual sense of Verdier, the category of modules A - ModD is a priori only a pre-tensor-triangulated category, but this does not cause any trouble ∼ for the definition. Since C is tensor-triangulated by assumption, the equivalence C = A - ModD just forces A - ModD to be tensor-triangulated as well. This technicality doesn’t arise when working in the 2-category of N-tensor-triangulated categories for any 2 ≤ N ≤ ∞. FINITE ETALE´ MORPHISMS IN TENSOR TRIANGULAR GEOMETRY 15 while the adjoint of the associated invariant form is obtained by going along the bottom. 4.7. Theorem. Let f ∗ : D → C be a geometric functor between rigidly-compactly generated tensor-triangulated categories. Then f ∗ is a finite étalemorphism (Defi- nition 4.1) if and only if the following three conditions hold: (a) f ∗ satisfies Grothendieck–Neeman duality; (b) the right adjoint f∗ is conservative; (c) the map 1C → ωf adjoint to the trace map is an isomorphism. Proof. (⇒) If f ∗ is finite étalethen it is extension-of-scalars with respect to the compact separable commutative algebra f∗(1). By the separable Neeman–Thomason Localization Theorem established by Balmer [Bal16a, Theorem 4.2], the compact objects in C are precisely the thick subcategory generated by the image f ∗(Dc) of the compact objects in D. Thus, by the projection formula, the fact that c ∗ f∗(1) is compact ensures that f∗(c) is compact for all c ∈ C . Thus, f satisfies Grothendieck–Neeman duality. The right adjoint f∗ is certainly conservative (in fact faithful). Moreover, by Corollary 2.38, the commutative rigid separable algebra f∗(1) is strongly separable, so that its trace form is nondegenerate. By Lemma 4.5, this means that the canonical map 1C → ωf becomes an isomorphism after applying f∗. But this means the canonical map is an isomorphism since f∗ is conservative. (⇐) If f ∗ satisfies Grothendieck–Neeman duality then the commutative algebra f∗(1C) is rigid hence has a trace map (so that part (c) makes sense). Moreover, by Lemma 4.5, if the map 1 → ωf adjoint to the trace map is an isomorphism then the trace form is nondegenerate; hence by Corollary 2.38, f∗(1C) is a (strongly) separable algebra. By Proposition 3.8, we have a tensor-triangulated equivalence C → f∗(1) - ModD compatible with the two adjunctions. Here we use the assumption that f∗ is conservative and the fact that C is idempotent complete (since it has ∗ small coproducts). Therefore f is finite étale.

4.8. Remark. Although in part (b) of Theorem 4.7 we just assume f∗ is conservative, it follows from the other hypotheses that it is actually faithful. It also follows from (a) and (c) that f ∗ has the full Wirthm¨ullerisomorphism of [BDS16, Theorem 1.9].

5. Examples We will now discuss some examples of finite étalemorphisms with an eye to future applications. 5.1. Example. Let G be a compact Lie group. It was proved in [BDS15, Theorem 1.1] that for any finite index subgroup H ≤ G, the restriction functor SH(G) → SH(H) between equivariant stable homotopy categories is finite étale. We can use Theo- rem 4.7 to improve this to an if and only if statement: 5.2. Theorem. Let G be a compact Lie group and let H ≤ G be a closed subgroup. G The restriction functor resH : SH(G) → SH(H) is finite étaleif and only if H has finite index in G. Proof. As already mentioned, the “if” part is [BDS15, Theorem 1.1]. For the “only G if” part recall that the relative dualizing object for resH is the representation sphere SL(H;G) for the tangent H-representation at the coset eH ∈ G/H (see [May03] and 16 BEREN SANDERS

G [San19, Remark 2.16]). By Theorem 4.7, if resH is finite étale,the canonical mor- L(H;G) phism 1SH(H) → S is an isomorphism. Restricting to the trivial subgroup, we obtain an isomorphism S0 → Sdim(G/H) in the nonequivariant stable homotopy category SH. The dimension of (the suspension spectrum of) a sphere is recovered by rational cohomology. Hence dim(G/H) = 0. The compact 0-dimensional mani- fold G/H is just a finite collection of points. That is, H has finite index in G. 1 1 n 5.3. Example. Let pn : S → S denote the degree n map z 7→ z on the unit circle. ∗ 1 1 The induced functor pn : SH(S ) → SH(S ) is not finite étale(for n ≥ 2). Indeed, 1 1 this amounts to the question of whether the quotient S → S /Cn by the subgroup 1 1 of nth roots of unity induces a finite étalemorphism SH(S /Cn) → SH(S ). But G [San19, Proposition 3.2] establishes that inflation inflG/N : SH(G/N) → SH(G) never satisfies Grothendieck–Neeman duality except when N = 1 is the trivial subgroup. 5.4. Remark. Another way of appreciating why Example 5.3 is not finite étaleis to look at its behaviour on the Balmer spectrum, which we know due to [BGH20, BS17]. The points of Spc(SH(S1)c) are of the form P(H, C) for H a closed subgroup of S1 and C ∈ Spc(SHc). The closed subgroups of S1 are, in addition to S1 itself, 1 the finite cyclic groups Cm (m ≥ 1) realized as the roots of unity in S . Consider the map on the Balmer spectrum ∗ 1 c 1 c ϕ := Spc(pn) : Spc(SH(S ) ) → Spc(SH(S ) ) 1 1 induced by the degree n map pn : S → S . One can show that ϕ(P(Cm, C)) = P(Clcm(m,n)/n, C). For example, taking n = 2 and fixing the nonequivariant prime C, it maps ( m/2 if 2 | m m 7→ m if 2 - m. In particular, we find that the fibers have cardinality ( −1 1 if 2 | N |ϕ ({P(CN , C)})| = 2 if 2 - N. For example, the fiber over P(C1, C) consists of two points P(C1, C), P(C2, C) . Moreover, if the nonequivariant prime C is 2-local then P(C1, C) ⊆ P(C2, C) is a nontrivial inclusion in the fiber. This implies that the basic theorems of Balmer [Bal16b, Theorem 1.5] on the behaviour of finite étalemorphisms do not hold for ∗ 1 1 the morphisms pn : SH(S ) → SH(S ). 5.5. Lemma. Consider a diagram of coproduct-preserving (N-)tensor-triangulated functors between rigidly-compactly generated (N-)tensor-triangulated categories

g∗ C D

h∗ k∗ C0 D0 f ∗ which commutes up to natural isomorphism of symmetric monoidal functors. De- ∗ ∗ note the right adjoints by f a f∗ and g a g∗ and suppose that the Beck–Chevalley comparison map ∗ ∗ h g∗ → f∗k FINITE ETALE´ MORPHISMS IN TENSOR TRIANGULAR GEOMETRY 17 is a natural isomorphism of lax symmetric monoidal functors. If g∗ is finite étale ∗ and f∗ is conservative then f is finite étale. ∗ ∗ Proof. The natural isomorphism h g∗ ' f∗k provides an isomorphism of commu- ∗ tative algebras h g∗(1D) ' f∗(1D0 ). By assumption, g∗(1D) is a compact commutative separable algebra in C, hence f∗(1D0 ) is a compact commutative separable 0 ∗ algebra in C . The f a f∗ adjunction satisfies the projection formula (see [BDS16, Prop. 2.15]) and f∗ is conservative by hypothesis. Hence, Proposition 3.8 provides the result. 5.6. Example. If C is a presentably symmetric monoidal stable ∞-category and A ∈ CAlg(C) is a commutative algebra in C, then we can consider the presentably symmetric monoidal stable ∞-category A - ModC of A-modules. If C is rigidly- compactly generated then so is A - ModC (see [PSW21, Remark 3.11], for example). At the level of homotopy categories, the extension-of-scalars Ho(C) → Ho(A - ModC) is then a geometric functor of rigidly-compactly generated ∞-tensor-triangulated categories whose right adjoint is conservative. 5.7. Example. Let C be a presentably symmetric monoidal stable ∞-category and let A, B ∈ CAlg(C) be commutative algebras in C. We then have

C B - ModC

A - ModC (A ⊗ B) - ModC where all four functors are extension-of-scalars. This is an example where the Beck– Chevalley property holds (at the level of the underlying stable ∞-categories). In particular, the induced diagram of ∞-tensor-triangulated categories

Ho(C) Ho(B - ModC)

Ho(A - ModC) Ho((A ⊗ B) - ModC) satisfies the first hypothesis of Lemma 5.5. Moreover, the right adjoints are all conservative (Example 5.6). Thus, if the top horizontal functor is finite étale(i.e. if B is a compact separable commutative algebra in Ho(C)) then the bottom horizontal functor is also finite étale.

5.8. Example. Let G be a compact Lie group and let SpG denote the symmetric monoidal stable ∞-category of G-spectra (see [GM20, Appendix C]). Let trivG : Sp → SpG denote the unique colimit-preserving symmetric monoidal functor from G the ∞-category of spectra. Since resH ◦ trivG ' trivH for any H ≤ G, we have a commutative diagram

Ho(SpG) Ho(SpH )

Ho(trivG(E) - ModSpG ) Ho(trivH (E) - ModSpH ) for any E ∈ CAlg(Sp). If H ≤ G has finite index then the top horizontal functor is finite étale(Example 5.1) and hence the bottom horizontal functor is finite étale. 18 BEREN SANDERS

Taking E = HZ, we obtain that the restriction functor

D(HZG) → D(HZH ) between categories of derived Mackey functors studied in [PSW21] is ﬁnite ´etale. This will be utilized in the forthcoming [BHS21] which will classify the localizing tensor-ideals of these categories. 5.9. Example. A version of Example 5.7 holds purely at the level of triangulated categories if one assumes the two algebras are separable. More precisely, let A and B be two commutative separable algebras in a rigidly-compactly generated N-tensor- triangulated category T. Iterated extension-of-scalars behaves as one expects (see [Pau17, Proposition 1.14]) and we have a diagram of rigidly-compactly generated N-tensor-triangulated categories

FA T A - ModT

B - ModT (A ⊗ B) - ModT which commutes up to isomorphism. Lemma 5.5 implies that if the top functor is finite étalethen so is the bottom functor. 5.10. Remark. Let F : D → C be a geometric functor of rigidly-compactly generated tensor-triangulated categories and let ϕ : Spc(Cc) → Spc(Dc) be the induced map on spectra. For any Thomason subset Y ⊆ Spc(Dc) with V := Spc(Dc) \ Y , we −1 have an induced functor F |V : D(V ) → C(ϕ (V )) on finite localizations such that D F C (5.11) F | D(V ) V C(ϕ−1(V )) commutes up to isomorphism. Moreover, on spectra Spc(F | ) ϕ−1(V ) =∼ Spc(C(ϕ−1(V ))) −−−−−−→V Spc(D(V )) =∼ V −1 is just the restriction ϕ|V : ϕ (V ) → V . 5.12. Example (Restriction in the target). If F : D → C is finite étalethen the induced “restriction” functor −1 F |V : D(V ) → C(ϕ (V )) of Remark 5.10 is also finite étale. Here V ⊆ Spc(Dc) is the complement of a Thomason subset. For example, V could be a quasi-compact open subset. Indeed this is just a special case of Example 5.9 with B = fV c the idempotent algebra for the finite localization D → D(V ). 5.13. Remark. Additional equivariant examples are featured in the work of Balmer and Dell’Ambrogio on Mackey 2-motives [BD20, Del21]. On the other hand, the following basic example relates the tensor-triangular notion of finite étalewith the ordinary scheme-theoretic notion: 5.14. Theorem (Balmer). If f : X → Y is a finite étalemorphism of quasi-compact ∗ and quasi-separated schemes then the derived functor Lf :Dqc(Y ) → Dqc(X) is a finite étalemorphism in the sense of Definition 4.1. FINITE ETALE´ MORPHISMS IN TENSOR TRIANGULAR GEOMETRY 19

Proof. This is provided by [Bal16a, Theorem 3.5]; see also [Nee18, Example 0.3]. 5.15. Remark. The proof of the above theorem works verbatim for other tensor- triangulated categories T(X) fibered over a category of schemes, provided the pseudofunctor X 7→ T(X) satisfies flat base change. Many motivic examples of such pseudofunctors are discussed in [CD19]. We just mention: 5.16. Example. Let L/K be a finite separable extension of fields whose characteristic (if positive) is invertible in the ring R. The induced functor SH(K; R) → SH(L; R) between motivic stable homotopy categories (with coefficients in R) is a finite étale morphism in the sense of Definition 4.1. The same is true of the induced functor DM(K; R) → DM(L; R) between derived categories of motives. See [CD19] and [Tot18] for more information about these categories. 5.17. Remark. The author thinks it is interesting to have an “intrinsic” characterization of finite étalemorphisms in tensor triangular geometry as expressed in Theorem 4.7. Nevertheless, actually classifying the finite étaleextensions of a given category T amounts to classifying the rigid (strongly) separable commutative algebras in T. For the equivariant stable homotopy category T = SH(G), this classification will be studied in forthcoming work with Balmer. The analogous problem for the stable module category T = StMod(kG) has been studied in [BC18] and is surprisingly subtle. It is currently only understood when G is cyclic.

5.18. Remark. For the derived category T = Dqc(X) of a noetherian scheme, Nee- man [Nee18] has obtained a very satisfactory classification of the (not necessarily compact) commutative separable algebras. His work shows that the tensor- triangular analogue of étalemorphism (a.k.a. extension by a commutative separable algebra) lies somewhere between the classical étalemorphisms of schemes and the pro-étalemorphisms of Bhatt–Scholze [BS15]. His results also show that there are no exotic étaleextensions of derived categories of schemes: An étaleextension of a derived category of a scheme is another derived category of a scheme. We’ll state this result precisely in the case of finite étaleextensions:

5.19. Theorem (Neeman). Let X be a noetherian scheme. If F :Dqc(X) → S is a finite étalemorphism (Def. 4.1) then there exists a finite étalemorphism of ∼ schemes f : U → X and a tensor-triangulated equivalence S = Dqc(U). With this ∗ identification, F is naturally isomorphic to Lf :Dqc(X) → Dqc(U). Proof. Let G denote the right adjoint of F . By definition, F is extension-of-scalars with respect to the compact commutative separable algebra G(1) ∈ Dqc(X). Nee- man [Nee18, Theorem 7.10] establishes that there is a separated finite-type étale map of schemes g : V → X and a generalization-closed subset U ⊂ V such that ∼ g G(1) = Rf∗(OU ) where f : U → X denotes the composite U,→ V −→ X. It then ∼ ∼ ∗ follows from Proposition 3.8 that S = Dqc(U) with F = Lf . Now, since G(1) is compact, the argument in [Nee18, Remark 0.6] shows that U ⊂ V is actually an ∗ ∗ open subset. (Take L := 0, Ke := f∗f (K) and the identity map Ke → f∗f K in loc. cit.) Thus, f : U → X is a separated finite-type étalemap. It is also proper since Lf ∗ =∼ F satisfies GN-duality (by [LN07]; see also [San19, Section 7] and [Lip09, Section 4.3]). This completes the proof since an étalemap is proper if and only if it is finite. 5.20. Remark. For the purpose of classifying the finite étaleextensions of a given tensor-triangulated category, the results of Section2 are worth keeping in mind. 20 BEREN SANDERS

They clarify that the compact/rigid commutative separable algebras that provide finite étaleextensions are necessarily self-dual. This puts limits on the role finite étalemorphisms can play in equivariant contexts over non-finite groups. Stated differently, Theorem 4.7 shows that the relative dualizing object ωf for a finite étalemorphism f ∗ must be trivial. It is natural to wonder if there is a reasonable generalization of “finite étale”in tensor triangular geometry which shares some of its good properties (e.g., the results of [Bal16a, Bal16b]) and yet covers examples having non-trivial dualizing objects (e.g., the examples which arise in [Rog08]).

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Beren Sanders, Mathematics Department, UC Santa Cruz, 95064 CA, USA Email address: [email protected] URL: http://people.ucsc.edu/∼beren/